Elements of the Representation Theory of the Jacobi Group

224 Pages·1998·2.108 MB·English

by  Rolf Berndt, Ralf Schmidt (auth.)

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Rolf Berndt
Ralf Schmidt
Elements of the Representation
Theory of the Jacobi Group
Reprint of the 1998 Edition
Rolf Berndt Ralf Schmidt
Department Mathematik Department of Mathematics
Universität Hamburg University of Oklahoma
Bundesstr. 55 Norman, OK 73019-3103
20146 Hamburg USA
Germany
ISBN978-3-0348-0282-6 e-ISBN978-3-0348-0283-3
DOI10.1007/978-3-0348-0283-3
Springer Basel Dordrecht Heidelberg London New York
Library of Congress Control Number: 2011941499
MathematicsSubjectClassification (2010): 11F55, 11F50, 11F70, 14K25, 22E50, 22E55
© Springer Basel AG 1998
Reprint of the 1st edition 1998 by Birkhäuser Verlag, Switzerland
Originally published as volume 163 in the Progress in Mathematics series

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AuffallenderWeisehateinesowichtigeFunctionnoch
keinen andern Namen, als den der Transcendente Θ,
nachderzufa¨lligenBezeichnung,mitdersiezuerstbei
Jacobi erscheint,unddieMathematikerwu¨rdennur
eine Pflicht der Dankbarkeit erfu¨llen, wenn sie sich
vereinigten ihr Jacobis Namen beizulegen, um das
AndenkendesMannes zuehren, zudessen sch¨onsten
Entdeckungen es geh¨ort, die innere Natur und hohe
Bedeutung dieser Transcendente zuerst erkannt zu
haben.
from: L. Dirichlet: Geda¨chtnisrede aufC.G.J. Jacobi
Preface
The Jacobigroupis a semidirectproduct of a symplectic groupwith a Heisen-
berg group. Its importance prima facie stems from the fact that it sets the
frame to treat theta functions and elliptic and abelian functions. Up to now,
mostworkconcerningthisgrouphasbeendoneforthesimplestcase“ofdegree
one”,wherethesymplecticgroupissimplySL(2)andtheHeisenberggroupisa
threeparameternilpotentgroup. TheJacobigroup,whosetheoryisintensively
interwoven with that of the metaplectic group, is, together with the Heisen-
berg group, the most evident example for a non-reductive group. This treatise
is meant to show how the general theory of automorphic forms for reductive
groups extends by some slight alterations to this first more general example.
The reader will see that a lot of the following may easily be extended to the
higher degree case of a semidirect product of a symplectic group Sp(n) with
a corresponding Heisenberg group. We were tempted to do this, but as the
generalizations are sometimes fairly easy on the one hand, and as the degree-
one case has special features, e.g. concerning the cusp conditions, on the other
hand, we restrict ourselves to this case, denoted GJ, here.
v

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 The Jacobi Group
1.1 Definition of GJ . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 GJ as an algebraic group . . . . . . . . . . . . . . . . . . . . . 3
1.3 The Lie algebra of GJ . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 GJ over the reals . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Basic Representation Theory of the Jacobi Group
2.1 Induced representations . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The Schro¨dinger representation . . . . . . . . . . . . . . . . . . 18
2.3 Mackey’s method for semidirect products . . . . . . . . . . . . 21
2.4 Representations of GJ with trivial central character . . . . . . 22
2.5 The Schro¨dinger-Weil representation . . . . . . . . . . . . . . . 24
2.6 Representations of GJ with non-trivial central character . . . . 28
3 Local Representations: The Real Case
3.1 Representations of gJ . . . . . . . . . . . . . . . . . . . . . . . 32
C
3.2 Models for infinitesimal representations and unitarizability. . . 39
3.3 Representations induced from BJ . . . . . . . . . . . . . . . . . 48
3.4 Representations induced from KJ and the automorphic factor . 51
3.5 Differential operators on X=H×C . . . . . . . . . . . . . . . 59
3.6 Representations induced from NˆJ and Whittaker models. . . . 63
4 The Space L2(ΓJ\GJ(R)) and its Decomposition
4.1 Jacobi forms and more general automorphic forms . . . . . . . 76
4.2 The cusp condition for GJ(R) . . . . . . . . . . . . . . . . . . . 83
4.3 The discrete part and the duality theorem . . . . . . . . . . . . 88
4.4 The continuous part . . . . . . . . . . . . . . . . . . . . . . . . 94
vii
viii Contents
5 Local Representations: The p-adic Case
5.1 Smooth and admissible representations . . . . . . . . . . . . . . 105
5.2 Whittaker models for the Schro¨dinger-Weil representation . . . 107
5.3 Representations of the metaplectic group. . . . . . . . . . . . . 112
5.4 Induced representations . . . . . . . . . . . . . . . . . . . . . . 115
5.5 Supercuspidal representations . . . . . . . . . . . . . . . . . . . 119
5.6 Intertwining operators . . . . . . . . . . . . . . . . . . . . . . . 121
5.7 Whittaker models. . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.8 Summary and Classification . . . . . . . . . . . . . . . . . . . . 132
5.9 Unitary representations . . . . . . . . . . . . . . . . . . . . . . 135
6 Spherical Representations
6.1 The Hecke algebra of the Jacobi group . . . . . . . . . . . . . . 138
6.2 Structure of the Hecke algebra in the good case . . . . . . . . . 140
6.3 Spherical representations in the good case . . . . . . . . . . . . 148
6.4 Spherical Whittaker functions . . . . . . . . . . . . . . . . . . . 153
6.5 Local factors and the spherical dual . . . . . . . . . . . . . . . 163
6.6 The Eichler-Zagieroperators . . . . . . . . . . . . . . . . . . . 167
7 Global Considerations
7.1 Adelization of GJ . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.2 The global Schro¨dinger-Weilrepresentation . . . . . . . . . . . 176
7.3 Automorphic representations . . . . . . . . . . . . . . . . . . . 179
7.4 Lifting of Jacobi forms . . . . . . . . . . . . . . . . . . . . . . . 183
7.5 The representation corresponding to a Jacobi form . . . . . . . 193
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Introduction
AfterPyatetski-Shapiro[PS1]andSatake[Sa1]introduced,independentofone
another, an early form of the Jacobi Theory in 1969 (while not naming it as
such), this theory was given a definite push by the book The Theory of Jacobi
Forms by Eichler and Zagier in 1985. Now, there are some overview articles
describingthedevelopmentsinthetheoryoftheJacobigroupanditsautomor-
phicforms,forinstancebySkoruppa[Sk2],Berndt[Be5]andKohnen[Ko]. We
refertotheseformorehistoricaldetailsandmanymorenamesofauthorsactive
inthistheory,whichstretchesnowfromnumbertheoryandalgebraicgeometry
totheoreticalphysics. Butletusonlybrieflyindicateseveral–sometimesvery
closely related – topics touched by Jacobi theory as we see it:
• fieldsofmeromorphicandrationalfunctionsontheuniversalellipticcurve
resp. universal abelian variety
• structure and projective embeddings of certain algebraic varieties and
homogeneous spaces
• correspondences between different kinds of modular forms
• L-functions associated to different kinds of modular forms and automor-
phic representations
• induced representations
• invariant differential operators
• structure of Hecke algebras
• determination of generalized Kac-Moody algebras
and as a final goal related to the here first mentioned
• mixed Shimura varieties and mixed motives.
Now, letting completely aside the arithmetical and algebraic geometrical ap-
proach to Jacobi forms developed and instrumentalized by Kramer [Kr], we
ix
x Introduction
will treat here a certain representation theoretic point of view for the Jacobi
theory parallel to the theory of Jacquet-Langlands [JL] for GL(2) as reported
by Godement [Go2], Gelbart [Ge1] and, recently, Bump [Bu]. Our text assem-
bles and regroups material from several papers mainly on the real theory by
the first-namedauthor,andbythe second-namedauthorsomedefinite amelio-
rations and additions to the non-archimedean and adelic theory contained in
the thesis of Homrighausen [Ho].
More precisely, our aim is
• to give a classificationof the irreducible unitary representations π of the
Jacobi group GJ over local fields,
• to construct explicit models for these representations, in particular the
Whittaker models,
• to discuss the relation between automorphic forms for GJ, i.e. the holo-
morphicJacobiforms,theirskewholomorphiccounterparts(firststudied
by Skoruppa) and possible generalizations, and the automorphic repre-
sentations of GJ, and
• topreparethegroundforafurtherdiscussionofautomorphicL-functions.
To reach these aims, we pursue the following plan.
InthefirstchapterwepresenttheJacobigroupGJ insomedifferentrealizations
and determine its Lie algebragJ. This gives some ideas about the structure of
our non-reductive GJ, and indicates in particular the important subgroups of
GJ one should look at. We take a closer look at the real points GJ(R) of GJ,
where we find a sort of generalized Iwasawa decomposition GJ = NJAJKJ.
Here NJ is a substitute for the unipotent radical of a maximal parabolic sub-
group in the reductive theory. It is characterizedas the closed connected sub-
group of GJ whose Lie algebra is the sum of positive root spaces.
In chapter two a method of Mackey will lead us to the fundamental principle
in the representation theory of GJ, which reads in our language:
π (cid:3)π˜⊗πm
SW
HereπisarepresentationofGJ,π˜isagenuinerepresentationofthemetaplectic
groupMp,andπm isacertainprojectivestandardrepresentationofGJ,called
SW
the Schro¨dinger-Weil representation. The meaning of the above equation is
that there is a 1-1 correspondence
(cid:2) (cid:3) (cid:4) (cid:5)
Irreducible representations
Irreducible, genuine re-
ofGJ withfixednon-trivial ←→ .
presentations of Mp
central character
One of our objectives in the later chapters is to make the above isomorphism
explicit, thereby showing that this bijection is canonical.